## Spacetime Diagrams II

Spacetime Diagrams II Spacetime Diagrams II Relativity and Astrophysics Lecture 22 Terry Herter Outline     Spacetime diagrams Worldlines Non-c...
Author: Alison Black
Spacetime Diagrams II

Spacetime Diagrams II Relativity and Astrophysics Lecture 22 Terry Herter

Outline    

Spacetime diagrams Worldlines Non-constant motion Twin paradox  

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Rotating lines of simultaneity Sending out a signal

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Spacetime Diagrams II

Spacetime diagram (or map)  

Plots space on one axis and time on the other Payoffs  

Place space and time on equal footing Review history of events and motions along a given line in spacetime Plot same events in different spacetime diagrams for different inertial reference frames  Can find what is different and what is the same between the

frames same place

same time

D

time B

Reference event O, along with other events A, B, C, and D. The dashed horizontal line represents events simultaneous in time while the dashed vertical line represents events at the same location.

C

A O

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space

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Same events, different frames 

Considers a light flash reflected off a mirror and back as seen from three different frames  

rocket, lab, and super-rocket The mirrors are at rest in the rocket frame and the superrocket is traveling in the same direction but faster (relative to lab frame) than the rocket

The light paths look like that shown below: rocket

lab

super-rocket

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Spacetime interval is conserved 

The spacetime interval is conserved (the same for each frame). (interval)2 = (time)2 – (space)2 = constant

The red hyperbola in the spacetime plots show the line of constant interval. The arrows represent the same arrow in spacetime! 

lab time

R

Maps show different perspectives of the same arrow in spacetime.

Spacetime geometry is Lorentz geometry, not Euclidean.

lab space

rocket time

super-rocket time R

R

super-rocket space

rocket space

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Worldline 

A worldline tracks the passage of a particle through spacetime

The worldline is a path in spacetime

 

 

Every particle has a worldline The spacetime map is an image of the worldline of a particle in a particular inertial frame Clocks don’t move (in x) in lab frame but move along in time – time passes at lattice points so that worldline of clocks are vertical

In the lab frame the line of simultaneity is horizontal Note: All particles have worldlines, even “stationary” ones. 5

1

2

1

2

3

4 5

4

3 Worldline of particles in spacetime for the lab frame. The world line for particle 1 contains a sample set of event points that make up the worldline.

time

O

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Trajectories of particles in space (not spacetime). Each particle starts at the reference clock (square) and moves with constant velocity.

space

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Non-constant speed particles 

Accelerated particles  

Have curved worldlines which represent their non-constant speed (acceleration is due to a force on the particle) This is true even if the particle is moving in a spatially straight line, like a train on a track.

Usefulness of spacetime diagrams – 

Useful for recognizing patterns of events for looking at the laws of nature but useless for influencing the events they represent (since events are recorded and sent back to the (ideal) observer) limits on worldline slope

time

Worldline of an accelerated particle. The particle stops at Z and then continues to P.

P

Z At every point in spacetime the particle is limited by the speed of light which place a limiting slope in the spacetime diagram.

O

space

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Length along a path 

Distance along a spatial path    

 

We could use a flexible tape or short stick to measure spatial distance along a track or path (Spatial distance)2 = (x2 - x1)2 + (y2 - y1)2 for each segment Sum together distances to get the total length The path lengths are the same for all surveyors

Different paths give different lengths Straight line 

Shortest path/distance between two points 14 10

The total length of the curved/winding path is long that the straight line between the two end points. The increased length for any segment is given by

12

8

10

6

8

north

 increase   increase   increase           in length   in north   in east  2

6

4 4

2 1/ 2

  

2 2 O

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east

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Proper time along a worldline 

Wristwatch (proper) time    

 

Sum spacetime interval along a worldline in Lorentz geometry the same way distance measures path length in Euclidean geometry (interval)2 = (t2 - t1)2 - (x2 - x1)2 for each segment Sum together the intervals to get the total proper time The proper time is the same for all inertial observers

Different world lines give different total proper times Straight worldline (free particle) 

Longest proper time between two points 8

The total proper time of the curved/winding worldline is shorter than the proper time for the straight worldline between the two end points. The increased proper time for any segment is given by

10 8

6

time

2  increase in   advance         proper time   in time 

6 4

4 2

2

O

2 1/ 2

  

Principle of Maximal Aging: a free particle follows the worldline of maximal aging

space

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 increment     in space 

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Principle of Maximal Aging 

A free particle follows the worldline of maximal aging

In Special Relativity straight worldlines have the longest proper time between two points

All inertial observers agree on which worldline is straight and has the longest proper time

Worldline in Spacetime

increase in east

north

direct path

O

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increase in space

time

increase in north

 increase  2    in north  

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B

Path in Space

B

2  increase       in east  

straight worldline 1/ 2

increase in time

 increase  2  increase  2      in space    in time    

 increase      in north 

east

O

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In spacetime the curved worldline is transversed in shorter proper time.

1/ 2

 increase      in time 

space

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“Kinked” Worldline For a curved worldline the clock associated with the particle is accelerating We take the limit in which we have a large acceleration over a short time

The worldline is straight except for a negligible part of the time when it changes direction

space = 0 m time = 10 m

B

(total proper time along OPB) = 10 meters of time

Zero proper time for light time

(proper time along leg OR)2 = (5 m)2 – (5 m)2 =0

space = 5 m time = 5 m Q

P

4

5

Reduced proper time along kinked worldline

space = 4 m time = 5 m

3

space = 0 m time = 0 m

(total proper time along leg OR)2 = 2x(proper time along OR ) =0

R

(proper time along leg OQ)2 = (5 m)2 – (4 m)2 = (3 m)2 (total proper time along leg OQB) = 2x(proper time along OQ ) =6m

4 O

space

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Sample Problem 5-1 

Consider a traveler who moves along the solid straight lines in the figure below.  

What is her proper time between the various events? What is here total proper time (aging) along the worldline 1, 2, 3, 4?

Her brother moves along the straight dotted line from event 1 to 4 directly.

Which twin is younger when they rejoin at event 4?

Calculate the increase in time on his clock 4

6 Event

Space (yr)

Time (yr)

1

1

0

2

1

1

3

-0.5

3

4

2

6

time

Kinked Path Event 1->2: (interval) = sqrt(12 – 02) = 1 yr 2->3: (interval) = sqrt(22 – 1.52) = 1.32 yr 3->4: (interval) = sqrt(32 – 2.52) = 1.66 yr

5 4

Total time:

3

Straight Path

2

Event 1->4: (interval) = sqrt(62 – 12)

2 1

-1

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0 1 space

= 5.92 yr

The brother (straight line path) ages more than the sister (kinked) path.

1

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= 1 + 1.32 + 1.66 = 3.98 yrs

3

2

3

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